Non-central Χ2 Research Articles

Robust Bayesian target vector optimization for multi-stage manufacturing processes

This research focuses on optimizing multi-stage manufacturing processes using Bayesian optimization (BO) with a robust Expected Improvement (EI) acquisition function. The aim is to optimize towards pre-selected target vectors, not to just minimize or maximize a function. To achieve this, we minimize the Euclidean distance between the actual and target output vectors, which requires transforming the Gaussian surrogate model posterior distribution into a non-central χ2(NCχ2) distribution. Furthermore, the distance measure additionally uses aleatoric uncertainty estimates of the actual output vectors to achieve robustness. We use a cascaded method that also considers the optimization results of intermediate stages, whereby optimization results are propagated from the last stage towards the first stage in each optimization iteration. By considering intermediate process outputs and aleatoric effects, our approach provides a robust optimization method for multi-stage manufacturing processes. To validate our method and to evaluate its properties, we use two artificial use cases. Moreover, we evaluate our approach in an industrial multi-stage forging process for the manufacturing of a nickel basis superalloy turbine disk, where involved stages are represented by corresponding 2D finite element method (FEM) DEFORM simulations. Evaluation suggests that our approach is superior in optimizing multi-stage manufacturing processes by considering all stage outcomes, robust distance measures, and the use of appropriate uncertainty distributions.

Approximating the Mode of the Non-Central Chi-Squared Distribution

In this paper we consider the probability density function (pdf) of the non-central χ2 distribution with arbitrary number of degrees of freedom and non-centrality. For this function we find the approximate location of the maximum and discuss related edge cases of 1 and 2 degrees of freedom. We also use this expression to demonstrate the improved performance of the C++ Boost’s implementation of the non-central χ2 and extend the domain of its applicability.

Higher Order AMMI (HO-AMMI) analysis: A novel stability model to study genotype-location interactions

Additive main effects and multiplicative interaction (AMMI) model is most widely used to analyze genotype x environment interactions (GEI) wherein interaction effects of location is masked by year effect. Hence, presently available models are not able to estimate interaction effects of genotype x location (GLI) and genotype x year (GYI) separately. Moreover, genotype ranking differs as number of years of evaluation vary making selection of genotype for target location difficult. In the present study, a novel stability model i.e., Higher-order-AMMI (HO-AMMI) analysis which can calculate GLI without the confounding effect of GYI and GLYI has been proposed. GEI of AMMI model and all 2-way interactions of HO-AMMI model follow χ2 distribution, whereas 3-way interaction (GLYI) of HO-AMMI follow noncentral χ2 distribution. With increase in number of years of evaluation contribution of GLI towards total variation increased whereas in AMMI model contribution of GEI towards total variation decreased. Variation explained by multiplicative components is higher in HO-AMMI compared to AMMI model. Genotypes were ranked using GL, GY and GL+GY+GLY interactions of HO-AMMI and GEI of AMMI for stability and yield and compared their ranks with field ranking. Correlation and linear regression analysis have indicated high association of GLI (HO-AMMI) with field ranking with high R2 values. Further, HO-AMMI model was able to remove the confounding effect of GYI and GLYI on GLI for accurate identification of genotype for target location irrespective of number of years of evaluation. Hence, HO-AMMI model can be used under multi-environment trials(MET) for selecting genotypes efficiently.

LINEAR HYPOTHESIS TESTING FOR HIGH DIMENSIONAL GENERALIZED LINEAR MODELS.

This paper is concerned with testing linear hypotheses in high-dimensional generalized linear models. To deal with linear hypotheses, we first propose constrained partial regularization method and study its statistical properties. We further introduce an algorithm for solving regularization problems with folded-concave penalty functions and linear constraints. To test linear hypotheses, we propose a partial penalized likelihood ratio test, a partial penalized score test and a partial penalized Wald test. We show that the limiting null distributions of these three test statistics are χ2 distribution with the same degrees of freedom, and under local alternatives, they asymptotically follow non-central χ2 distributions with the same degrees of freedom and noncentral parameter, provided the number of parameters involved in the test hypothesis grows to ∞ at a certain rate. Simulation studies are conducted to examine the finite sample performance of the proposed tests. Empirical analysis of a real data example is used to illustrate the proposed testing procedures.

Universal recurrence algorithm for computing Nuttall, generalized Marcum and incomplete Toronto functions and moments of a noncentral χ2 random variable

This article proposes a novel recurrence algorithm for computing Nuttall, generalized Marcum, and incomplete Toronto functions as well as truncated and raw moments (of any real order) for a noncentral χ2 random variable. The numerical results highlight that the proposed recurrence method offers an excellent speed-accuracy trade-off for a wide variety of meaningful and relevant applications ranging from the operations management and industrial engineering fields to problems encountered in the financial engineering industry.

Confidence Intervals of Fit Indexes by Inverting a Bootstrap Test

Fit indexes are an important tool in the evaluation of model fit in structural equation modeling (SEM). Currently, the newest confidence interval (CI) for fit indexes proposed by Zhang and Savalei (2016) is based on the quantiles of a bootstrap sampling distribution at a single level of misspecification. This method, despite a great improvement over naive and model-based bootstrap methods, still suffers from unsatisfactory coverage. In this work, we propose a new method of constructing bootstrap CIs for various fit indexes. This method directly inverts a bootstrap test and produces a CI that involves levels of misspecification that would not be rejected in a bootstrap test. Similar in rationale to a parametric CI of root mean square error of approximation (RMSEA) based on a noncentral χ2 distribution and a profile-likelihood CI of model parameters, this approach is shown to have better performance than the approach of Zhang and Savalei (2016), with more accurate coverage and more efficient widths.

On the statistical power of Baarda’s outlier test and some alternative

Abstract Baarda’s outlier test is one of the best established theories in geodetic practice. The optimal test statistic of the local model test for a single outlier is known as the normalized residual. Also other model disturbances can be detected and identified with this test. It enjoys the property of being a uniformly most powerful invariant (UMPI) test, but is not a uniformly most powerful (UMP) test. In this contribution we will prove that in the class of test statistics following a common central or non-central χ2 distribution, Baarda’s solution is also uniformly most powerful, i.e. UMPχ2 for short. It turns out that UMPχ2 is identical to UMPI, such that this proof can be seen as another proof of the UMPI property of the test. We demonstrate by an example that by means of the Monte Carlo method it is even possible to construct test statistics, which are regionally more powerful than Baarda’s solution. They follow a so-called generalized χ2 distribution. Due to high computational costs we do not yet propose this as a ”new outlier detection method”, but only as a proof that it is in principle possible to outperform the statistical power of Baarda’s test.

Monotonicity properties and bounds for the chi-square and gamma distributions

The generalized Marcum functions Qμ(x,y) and Pμ(x,y) have as particular cases the non-central χ2 and gamma cumulative distributions, which become central distributions (incomplete gamma function ratios) when the non-centrality parameter x is set to zero. We analyze monotonicity and convexity properties for the generalized Marcum functions and for ratios of Marcum functions of consecutive parameters (differing in one unity) and we obtain upper and lower bounds for the Marcum functions. These bounds are proven to be sharper than previous estimations for a wide range of the parameters. Additionally we show how to build convergent sequences of upper and lower bounds. The particularization to incomplete gamma functions, together with some additional bounds obtained for this particular case, lead to combined bounds which improve previously existing inequalities.

Adaptive test statistics for ergodic diffusion processes sampled at discrete times

We consider adaptive test statistics for ergodic diffusion processes based on discrete observations. Since an exact likelihood function for the discretely observed diffusion process cannot been generally obtained, the quasi-likelihood function based on the Ito–Taylor expansion is used and three kinds of test statistics, likelihood ratio type test statistic, Wald type test statistic and Rao׳s score type test statistic, for diffusion processes are proposed. It is shown that the test statistics converge in distribution to χ2 (the chi-squared distribution) under null hypothesis and the tests are consistent. Moreover, we prove that the test statistics converge in distribution to the noncentral χ2 under local alternative hypothesis.

Sample size calculation for comparing time-averaged responses in [formula omitted]-group repeated-measurement studies

Many clinical trials compare the efficacy of K (≥3) treatments in repeated measurement studies. However, the design of such trials has received relatively less attention from researchers. Zhang and Ahn (2012) derived a closed-form sample size formula for two-sample comparisons of time-averaged responses using the generalized estimating equation (GEE) approach, which takes into account different correlation structures and missing data patterns. In this paper, we extend the sample size formula to scenarios where K (≥3) treatments are compared simultaneously to detect time-averaged differences in treatment effect. A closed-form sample size formula based on the noncentral χ2 test statistic is derived. We conduct simulation studies to assess the performance of the proposed sample size formula under various correlation structures from a damped exponential family, random and monotone missing patterns, and different observation probabilities. Simulation studies show that empirical powers and type I errors are close to their nominal levels. The proposed sample size formula is illustrated using a real clinical trial example.

Testing Linearity in Cointegrating Relations With an Application to Purchasing Power Parity

This article shows that when applied to nonstationary time series, the conventional Regression Error Specification Test (RESET) leads to severe size distortion and its asymptotic distribution involves a mixture of noncentral χ2 distributions. Nonstationarity introduces bias terms in the limit distribution, and appropriate corrections for the bias are presented leading to a modified RESET test that has a central χ2 limit distribution. In simulations, this modified test is shown to have power not only against nonlinear cointegration but also against the absence of cointegration. In an empirical illustration, the linear purchasing power parity (PPP) specification is tested using five Organization for Economic Cooperation and Development (OECD) countries.

Empirical Bayes block shrinkage of wavelet coefficients via the noncentral χ2 distribution

Empirical Bayes approaches to the shrinkage of empirical wavelet coefficients have generated considerable interest in recent years. Much of the work to date has focussed on shrinkage of individual wavelet coefficients in isolation. In this paper we propose an empirical Bayes approach to simultaneous shrinkage of wavelet coefficients in a block, based on the block sum of squares. Our approach exploits a useful identity satisfied by the noncentral Χ 2 density and provides some tractable Bayesian block shrinkage procedures. Our numerical results indicate that the new procedures perform very well.

A score test under a semiparametric finite mixture model

We propose a score statistic to test the mixing proportion under a semiparametric finite mixture model introduced by Anderson [Anderson, J.A., 1979, Multivariate logistic compounds. Biometrika, 66, 17–26]. The proposed score test is based on the semiparametric profile log-likelihood function under a three-sample semiparametric model. The proposed score statistic has an asymptotic χ2 distribution under the null hypothesis and an asymptotic noncentral χ2 distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to the Wald test and the empirical likelihood ratio test. We present some results on simulation and on the analysis of one data set.

The S-system computation of non-central gamma distribution

The non-central gamma distribution can be regarded as a general form of non-central χ2 distributions whose computations were thoroughly investigated (Ruben, H., 1974, Non-central chi-square and gamma revisited. Communications in Statistics, 3(7), 607–633; Knüsel, L., 1986, Computation of the chi-square and Poisson distribution. SIAM Journal on Scientific and Statistical Computing, 7, 1022–1036; Voit, E.O. and Rust, P.F., 1987, Noncentral chi-square distributions computed by S-system differential equations. Proceedings of the Statistical Computing Section, ASA, pp. 118–121; Rust, P.F. and Voit, E.O., 1990, Statistical densities, cumulatives, quantiles, and power obtained by S-systems differential equations. Journal of the American Statistical Association, 85, 572–578; Chattamvelli, R., 1994, Another derivation of two algorithms for the noncentral χ2 and F distributions. Journal of Statistical Computation and Simulation, 49, 207–214; Johnson, N.J., Kotz, S. and Balakrishnan, N., 1995, Continuous Univariate Distributions, Vol. 2 (2nd edn) (New York: Wiley). Both distributional function forms are usually in terms of weighted infinite series of the central one. The ad hoc approximations to cumulative probabilities of non-central gamma were extended or discussed by Chattamvelli, Knüsel and Bablok (Knüsel, L. and Bablok, B., 1996, Computation of the noncentral gamma distribution. SIAM Journal on Scientific Computing, 17, 1224–1231), and Ruben (Ruben, H., 1974, Non-central chi-square and gamma revisited. Communications in Statistics, 3(7), 607–633). However, they did not implement and demonstrate proposed numerical procedures. Approximations to non-central densities and quantiles are not available. In addition, its S-system formulation has not been derived. Here, approximations to cumulative probabilities, density, and quantiles based on the method of Knüsel and Bablok are derived and implemented in R codes. Furthermore, two alternate S-system forms are recast on the basis of techniques of Savageau and Voit (Savageau, M.A. and Voit, E.O., 1987, Recasting nonlinear differential equations as S-systems: A canonical nonlinear form. Mathematical Biosciences, 87, 83–115) as well as Chen (Chen, Z.-Y., 2003, Computing the distribution of the squared sample multiple correlation coefficient with S-Systems. Communications in Statistics—Simulation and Computation, 32(3), 873–898.) and Chen and Chou (Chen, Z.-Y. and Chou, Y.-C., 2000, Computing the noncentral beta distribution with S-system. Computational Statistics and Data Analysis, 33, 343–360.). Statistical densities, cumulative probabilities, quantiles can be evaluated by only one numerical solver power low analysis and simulation (PLAS). With the newly derived S-systems of non-central gamma, the specialized non-central χ2 distributions are demonstrated under five cases in the same three situations studied by Rust and Voit. Both numerical values in pairs are almost equal. Based on these, nine cases in three similar situations are designed for demonstration and evaluation. In addition, exact values in finite significant digits are provided for comparison. Demonstrations are conducted by R package and PLAS solver in the same PC system. By doing these, very accurate and consistent numerical results are obtained by three methods in two groups. On the other hand, these three methods are performed competitively with respect to speed of computation. Numerical advantages of S-systems over the ad hoc approximation and related properties are also discussed.

Formulation of the EM algorithm for a mixture of central and non-central χ2 distributions

The expectation–maximization (EM) algorithm is applied to a mixture of central and non-central χ 2 distributions. The derivation is based on the writing of this mixture as an infinite mixture of exponential distributions for which the EM algorithm takes a simple form.

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function Iν and which are related to the noncentral χ2 -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).

Evaluation of the extent of nonlinearity in reportable range studies.

Abstract Objectives.—To extend the polynomial method for evaluating linearity in 2 ways. First, we developed a screen to ascertain whether the data were precise enough to permit a reliable evaluation of linearity and therefore eliminate findings of linearity due to low statistical power. Second, we assessed whether the degree of nonlinearity detected by the polynomial method was clinically relevant using a statistically rigorous method. Methods.—Because we assessed linearity relative to a clinically determined level of importance instead of the default value of zero, we used sampling theory based on the noncentral χ2 distribution. Using statistical power calculations, we incorporated a screen for imprecision that guarantees that the probability of correctly identifying nonlinear methods is at least 80%. Results.—With the described methods, we achieved a sensitivity of at least 80% and a specificity of at least 95%. When the data were too imprecise to achieve a sensitivity of 80%, no determination of linea...

Evaluation of the extent of nonlinearity in reportable range studies.

To extend the polynomial method for evaluating linearity in 2 ways. First, we developed a screen to ascertain whether the data were precise enough to permit a reliable evaluation of linearity and therefore eliminate findings of linearity due to low statistical power. Second, we assessed whether the degree of nonlinearity detected by the polynomial method was clinically relevant using a statistically rigorous method. Because we assessed linearity relative to a clinically determined level of importance instead of the default value of zero, we used sampling theory based on the noncentral chi(2) distribution. Using statistical power calculations, we incorporated a screen for imprecision that guarantees that the probability of correctly identifying nonlinear methods is at least 80%. With the described methods, we achieved a sensitivity of at least 80% and a specificity of at least 95%. When the data were too imprecise to achieve a sensitivity of 80%, no determination of linearity was made. This procedure mimics the practice in manual inspection of flagging data that appear imprecise by visual inspection and halting the evaluation. Formal statistical tests for precision and amount of nonlinearity are advantageous because they allow us to quantify and limit classification errors. By formalizing these various aspects of linearity assessment, we maintain some of the complex features of manual methods while making the linearity assessment feasible to apply to a high volume of assessments and removing the between-analyst variability.

A Wiener Germ approximation of the noncentral chi square distribution and of its quantiles

The cumulative distribution function (cdf) of the noncentral χ2 distribution with positive degrees of freedom ν > 0 and a noncentrality parameter δ2 ≥ 0 is usually expressed as an infinite weighted sum of central χ2 cdf’s. For the purpose of numerical evaluation this infinite sum is being approximated by a finite sum. For large values of the noncentrality parameter, the sum converges slowly. Alternative approximation algorithms have been proposed instead in the literature. A comparison of these is given in Johnson & Kotz (1970). Most of the approximation algorithms have advantages for certain values of the arguments/parameters and perform poorly for other values. We are proposing an approximation algorithm that has a very solid theoretical background and is surprisingly accurate for extremely large set of arguments/parameter values. It is also applied for a reliable approximation of the quantiles of the distribution for large values of noncentrality and degrees of freedom. Although being asymptotic in spirit (with respect to degrees of freedom ν), the algorithm gives quite accurate approximation even down to ν = 1.

Generalization for the distribution of hartley’S and nair’s Fmax statistics

The exact distribution of Hartley’s FmaxH statistics under heterogeneity of variances with or without unequal sample sizes is given by Yanagisawa and Shirakawa (1997). The distribution of Hartley’s FmaxH statistics is further generalized to that when follows a non-central χ2 distribution instead of a central χ2 distribution. The exact distribution of Nair’s FmaxN statistics under heterogeneity of variances with or without unequal sample sizes when follows either a central χ2 or a non-central χ2 distribution is given. These exact distributions provide new statistical tests and the power of the tests under very general conditions, such as a simultaneous statistical test for the Analysis of Variance (ANOVA). For these tests, we provide a Fortran program1 which calculates, to a high accuracy, the upper tail probabilities, probability points and probability densities for FmaxH and FmaxN statistics under very general conditions. Usually the accuracy of the calculation is 13 to 14 digits for double precision arithmetic. The probability density functions under various conditions are shown and an extension statistical table for Nair’s FmaxN statistic is given when the degrees of freedom is small.